An implicit finite difference scheme for the nonlinear size-structured population model

In this paper we consider the "fully nonlinear" size structured population model. We develop an implicit finite difference scheme to approximate the solution of this nonlinear partial differential equation. The convergence of this approximation to a unique bounded variation solution of this model is obtained. Numerical results to an example problem are presented.     

Parallel simulation of individual-based, physiologically structured population models

A general scheme for parallel simulation of individual-based, structured population models is proposed. Algorithms are developed to simulate such models in a parallel computing environment. The simulation model consists of an individual model and a population model that incorporates the individual dynamics. The individual model is a continuous time representation of organism life history for growth with discrete allocations for reproductive processes. The population model is a continuous time simulation of a nonlinear partial differential equation of extended McKendrick-von Foerster-type.

As a prototypical example, we show that a specific individual-based, physiologically structured model for Daphnia populations is well suited for parallelization, and significant speed-ups can be obtained by using efficient algorithms developed along our general scheme. Because the parallel algorithms are applicable to generic structured populations which are the foundation for populations in a more complex community or food-web model, parallel computation appears to be a valuable tool for ecological modeling and simulation.     

A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment

A numerical method is developed for a general structured population model coupled with the environment dynamics over a bounded domain where the individual growth rate changes sign. Sign changes notably exhibit nonlocal dependence on the population density and environmental factors (e.g., resource availability and other habitat variables). This leads to a highly nonlinear PDE describing the time‐evolution of the population density coupled with a nonlinear‐nonlocal system of ODEs describing the environmental time‐dynamics. Stability of the finite‐difference numerical scheme and its convergence to the unique weak solution are proved. Numerical experiments are provided to demonstrate the performance of the finite difference scheme and to illustrate a range of biologically relevant potential applications.     

A high resolution finite difference method for a model of structured susceptible‐infected populations coupled with the environment

We develop a general model describing a structured susceptible‐infected (SI) population coupled with the environment. This model applies to problems arising in ecology, epidemiology, and cell biology. The model consists of a system of quasilinear hyperbolic partial differential equations coupled with a system of nonlinear ordinary differential equations that represents the environment. We develop a second‐order high resolution finite difference scheme to numerically solve the model. Convergence of this scheme to a weak solution with bounded total variation is proved. Numerical simulations are provided to demonstrate the high‐resolution property of the scheme and an application to a multi‐host wildlife disease model is explored.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1420–1458, 2017     

Numerical analysis to discontinuous Galerkin methods for the age structured population model of marine invertebrates

In this article we consider the age structured population growth model of marine invertebrates. The problem is a nonlinear coupled system of the age‐density distribution of sessile adults and the abundance of larvae. We propose the semidiscrete and fully‐discrete discontinuous Galerkin schemes to the nonlinear problem. The DG method is well suited to approximate the local behavior of the problem and to easily take the locally refined meshes with hanging nodes adaptively. The simple communication pattern between elements makes the DG method ideal for parallel computation. The global existence of the approximation solution is proved for the nonlinear approximation system by using the broken Sobolev spaces and the Schauder's fixed point theorem, and error estimates are obtained for both the semidiscrete scheme and the fully‐discrete scheme. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Global weak solution for a multistage physiologically structured population model with resource interaction

We construct a multistage kinetic model of a physiologically structured insect population whose life history consists of fourth stages of development termed eggs, larval, pupal and adult moth (male and female). The model is a system of weakly coupled hyperbolic partial differential equations with nonlocal boundary conditions. The vital rates depend on the resource which satisfy an ordinary differential equation. We discretize the physiological space and formulate an implicit scheme and we prove the existence and uniqueness of the solution. The numerical simulation provides an analytical tool to improve the understanding of the moth’s biology.     

A moving boundary problem with space-fractional diffusion logistic population model and density-dependent dispersal rate

In this article, we discuss a space-fractional diffusion logistic population model with Caputo fractional derivative and density-dependent dispersal rate. The numerical solution of the problem is obtained by using a finite difference scheme. The consistency and stability of the scheme for our solution to the problem are also discussed. The effect of the density-dependent dispersal rate and order of the space-fractional derivative are analyzed for the population density and expanding front (moving boundary).     

Analytical and numerical investigations of a batch crystallization model

This article is concerned with the analytical and numerical investigations of a one-dimensional population balance model for batch crystallization processes. We start with a one-dimensional batch crystallization model and prove the local existence and uniqueness of the solution of this model. For this purpose Laplace transformation is used as a basic tool. A semi-discrete high resolution finite volume scheme is proposed for the numerical solution of the current model. The issues of positivity (monotonicity), consistency, stability and convergence of the proposed scheme for the current model are analyzed and proved. Finally, we give a numerical test problem. The numerical results of the proposed high resolution scheme are compared with the solution of the reduced four-moments model and the first-order upwind scheme.     

Chaotic effects on disease spread in simple eco-epidemiological system

In this paper, an eco-epidemiological model where prey disease is structured as a susceptible-infected model is investigated. Thresholds that control disease spread and population persistence are obtained. Existence, stability and instability of the system are studied. Hopf bifurcation is shown to occur where a periodic solution bifurcates from the coexistence equilibrium. Simulations show that the system exhibits chaotic phenomena when the transmission rate is varied.     

Dirac Concentrations in a Chemostat Model of Adaptive Evolution

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations.Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection.In this paper,the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed.Using either weak or strong regularity assumptions,the authors study the concentration of the solution.To this end,BV estimates in time on appropriate quantities are stated,and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.     

Dirac Concentrations in a Chemostat Model of Adaptive Evolution(In honor of the immense scientific influence of Ham Brezis)

This paper deals with a non-local parabolic equation of Lotka-Volterra type that describes the evolution of phenotypically structured populations. Nonlinearities appear in these systems to model interactions and competition phenomena leading to selection. In this paper, the equation on the structured population is coupled with a differential equation on the nutrient concentration that changes as the total population varies.Different methods aimed at showing the convergence of the solutions to a moving Dirac mass are reviewed. Using either weak or strong regularity assumptions, the authors study the concentration of the solution. To this end, BV estimates in time on appropriate quantities are stated, and a constrained Hamilton-Jacobi equation to identify where the solutions concentrates as Dirac masses is derived.     

一个改进的三阶WENO-Z型格式

在WENO-Z型格式框架下,基于高阶全局光滑因子,在非线性权建立过程中引入参数,通过收敛性分析确定参数取值范围,兼顾精确性与不振荡性,得到参数最佳取值.最终得到一个低耗散、高分辨率的三阶WENO差分格式,该格式在函数一阶极值点处仍保持预期三阶精度.最后通过精确解算例验证了格式在各种类型极值点处精度恢复情况,并通过一、二维Euler方程组经典算例测试了格式的低耗散、高分辨特性.结果表明,该文格式是一个性能优良的激波捕捉格式.     

Global Asymptotic Stability for a Class of Nonlinear Populations Model

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Nonlinear semigroup approach to age structured proliferating cell population with inherited cycle length

This paper deals with a nonlinear semigroup approach to semilinear initial-boundary value problems which model nonlinear age structured proliferating cell population dynamics. The model involves age-dependence and cell cycle length, and boundary conditions may contain compositions of nonlinear functions and trace of solutions. Hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. A family of equivalent norms is introduced to discuss local quasidissipativity of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators. The resultant solution operators are obtained as nonlinear semigroups which are not quasicontractive but locally equi-Lipschitz continuous.     

Turbulent compressible fluid flow computation in a male rotor-housinggap of screw compressors

This contribution is devoted to a turbulent compressible fluid flow computation through a 2-D model of the male rotor-housing gap in screw compressors. For the numerical solution of the nonlinear conservative system of the Favre-averaged Navier-Stokes equations the cell-centred finite volume formulation of the explicit two-step TVD MacCormack scheme proposed by Causon on a structured quadrilateral grid is used. The turbulent viscosity is calculated by using the algebraic Baldwin-Lomax turbulence model which is implemented into the own developed numerical code. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimal harvesting for nonlinear size-structured population dynamics

A harvesting problem is considered for a size structured population model with separable mortality rate and nonlinear fertility rate. We transform the model to a system of equations and show the existence of a unique solution. We also establish the existence of an optimal harvesting rate which maximizes the total harvest in a given time interval.     

Global boundedness of the solutions to a Gurtin-MacCamy system

This paper is concerned with the analysis of a generalized Gurtin-MacCamy model describing the evolution of an age-structured population. The problem of global boundedness is studied. Namely we ask whether there are simple general assumptions that one can make on the vital rates in order to have boundedness of the solution. Next a fully implicit finite difference scheme along the characteristic is considered to approximate the solution of the system. Global boundedness of the numerical solutions is investigated. The optimal rate of convergence of the scheme is obtained in the maximum norm. Numerical examples are presented.     

Continuous dependence results for a general model of size-dependent population dynamics

We are concerned with a general model of size structured population dynamics with the growth rate depending on the individual's size and time. In this paper, we shall study the continuous dependence of the solution on all given data such as aging and birth functions, growth rate functions and initial data.     

Approximation of a population dynamics model by parabolic regularization

The basic linear model for describing an age structured population spreading in a limited habitat is considered with the purpose of investigating an approximation procedure based on parabolic regularization. In fact, a viscosity model is introduced by considering an appropriate approximating regularized parabolic problem and it is proved that the sequence of the approximating solutions tends to the solution to the original problem. The advantage of this approach is that it leads to the numerical solution of a parabolic problem that has more stable solutions than the hyperbolic‐parabolic original problem and avoids the restrictions (compatibility conditions) needed to treat the latter. Moreover, for the solution of the approximating problem, it is possible to take advantage of established software packages dedicated to parabolic problems. Some examples of the approach are provided using COMSOL Multiphysics. Copyright © 2012 John Wiley & Sons, Ltd.     

Mathematical modelling of compressible viscous fluid flow in a male rotor‐housing gap of screw machines

This contribution is devoted to the mathematical modelling of a compressible viscous fluid flow through a 2‐D model of the male rotor‐housing gap in screw machines. Numerical solution of the nonlinear conservative system of the compressible Navier‐Stokes equations is obtained by means of the cell‐centred finite volume formulation of the explicit two‐step TVD MacCormack scheme proposed by Causon on a structured quadrilateral grid using the own developed numerical code. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)